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Theorem eqoreldif 4257
Description: An element of a set is either equal to another element of the set or a member of the difference of the set and the singleton containing the other element. (Contributed by AV, 25-Aug-2020.) (Proof shortened by JJ, 23-Jul-2021.)
Assertion
Ref Expression
eqoreldif (𝐵𝐶 → (𝐴𝐶 ↔ (𝐴 = 𝐵𝐴 ∈ (𝐶 ∖ {𝐵}))))

Proof of Theorem eqoreldif
StepHypRef Expression
1 simpl 472 . . . . 5 ((𝐴𝐶 ∧ ¬ 𝐴 = 𝐵) → 𝐴𝐶)
2 elsni 4227 . . . . . . 7 (𝐴 ∈ {𝐵} → 𝐴 = 𝐵)
32con3i 150 . . . . . 6 𝐴 = 𝐵 → ¬ 𝐴 ∈ {𝐵})
43adantl 481 . . . . 5 ((𝐴𝐶 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴 ∈ {𝐵})
51, 4eldifd 3618 . . . 4 ((𝐴𝐶 ∧ ¬ 𝐴 = 𝐵) → 𝐴 ∈ (𝐶 ∖ {𝐵}))
65ex 449 . . 3 (𝐴𝐶 → (¬ 𝐴 = 𝐵𝐴 ∈ (𝐶 ∖ {𝐵})))
76orrd 392 . 2 (𝐴𝐶 → (𝐴 = 𝐵𝐴 ∈ (𝐶 ∖ {𝐵})))
8 eleq1a 2725 . . 3 (𝐵𝐶 → (𝐴 = 𝐵𝐴𝐶))
9 eldifi 3765 . . . 4 (𝐴 ∈ (𝐶 ∖ {𝐵}) → 𝐴𝐶)
109a1i 11 . . 3 (𝐵𝐶 → (𝐴 ∈ (𝐶 ∖ {𝐵}) → 𝐴𝐶))
118, 10jaod 394 . 2 (𝐵𝐶 → ((𝐴 = 𝐵𝐴 ∈ (𝐶 ∖ {𝐵})) → 𝐴𝐶))
127, 11impbid2 216 1 (𝐵𝐶 → (𝐴𝐶 ↔ (𝐴 = 𝐵𝐴 ∈ (𝐶 ∖ {𝐵}))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wcel 2030  cdif 3604  {csn 4210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-dif 3610  df-sn 4211
This theorem is referenced by:  lcmfunsnlem2  15400
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